Future Higgs Studies: A Theorist's Outlook

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by  Howard E. Haber

Future Higgs Studies: A Theorist’s Outlook
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HowardE.Haber
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SantaCruzInstituteforParticlePhysics
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UniversityofCalifornia,SantaCruz,CA95064USA
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E-mail: [email protected]
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WeexaminesomeofthetheoreticalandphenomenologicalimplicationsoftheHiggsbosondis-
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covery and discuss what these imply for future Higgs studies at the LHC and future colliders.
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v Inparticular,oneoftheoutstandingunansweredquestionsiswhetheradditionalscalarsbeyond
2 theobservedHiggsbosonarepresentinthespectrumoffundamentalparticles. Anytheoryofa
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9 non-minimalHiggssectormustpossessa scalarstate whosepropertiesareapproximatelythose
1 oftheStandardModelHiggsboson. Thiscanbeachievedintheso-calledalignmentlimitofthe
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. extended scalar sector. Examples of scalar sectors in which an approximate alignment limit is
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0 achievedaresurveyed.
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ProspectsforChargedHiggsDiscoveryatColliders
3-6October2016
Uppsala,Sweden
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
1. Introduction
The discovery of the Higgs boson in 2012 [1,2] marked an important milestone in the study
of fundamental particles and their interactions. The Standard Model (SM) of particle physics is
now complete. Indeed, there are no definitive departures from the Standard Model observed in
experiments conducted at high energy collider facilities. Nevertheless, some fundamental micro-
scopicphenomenamustnecessarily lieoutsideofthepurviewoftheSM.Theseinclude: neutrinos
withnon-zeromass[3];darkmatter[4];thebaryonasymmetryoftheuniverse[5];thesuppression
of CP-violation in the strong interactions (the so-called strong CP problem [6]); gauge coupling
unification[7];inflationintheearlyuniverse[8];darkenergy[9];andthegravitational interaction.
Noneofthesephenomena canbeexplained withintheframeworkoftheSMalone.
As a result, the SM should be regarded at best as a low-energy effective field theory, which
is valid below somehigh energy scale L . Forexample, acredible theory of neutrino masses (e.g.,
the type-I seesaw model [3]) posits the existence of a right-handed electroweak singlet Majorana
neutrino of mass of order 1014 GeV. The gravitational force is governed by Planck-scale physics
corresponding to L 1019 GeV. Henceforth, we shall define L to be the lowest energy scale at
∼
which the SM breaks down. The predictions made by the SM depend on a number of parameters
thatmustbetakenasinputtothetheory. Theseparametersaresensitivetoultraviolet(UV)physics,
andsincethephysicsatveryhighenergiesisnotknown,onecannotpredicttheirvalues. Ingeneral,
fermions and boson masses depend differently on L [10]. On the one hand, fermion masses are
logarithmically sensitive to UV physics, due to the chiral symmetry of massless fermions, i.e.
d m m ln(L 2/m2). In contrast, no such symmetry exists to protect masses of spin-0 bosons
F ∼ f F
(in the absence of supersymmetry), and consequently weexpect quadratic sensitivity ofthe scalar
bosonsquared masstoUVphysics, d m2 L 2.
B∼
IntheSM,theHiggsscalarpotential,
V(F )= m 2(F †F )+1l (F †F )2, (1.1)
− 2
where m 2 = 1l v2 depends on the vacuum expectation value (vev) v of the Higgs field. The pa-
2
rameter m 2 isquadratically sensitive toL . Hence,toobtain v 246GeVinatheorywherev L
≃ ≪
requires a significant fine-tuning of the ultraviolet parameters of the fundamental theory. Indeed,
the one-loop contribution to the squared mass parameter m 2 would be expected to be of order
(g2/16p 2)L 2. Settingthisquantitytobeoforderofv2 (toavoidanunnaturalcancellationbetween
thetree-levelparameterandtheloopcorrections)yieldsL 4p v/g O(1TeV). Anaturaltheory
≃ ∼
of electroweak symmetry breaking (EWSB) would seem to require new physics at the TeV scale
associated withtheEWSBdynamics.
There have been anumber oftheoretical proposals to explain the origin of the EWSBenergy
scale: (1)naturalnessisrestoredbysupersymmetrywhichtiesthebosonstothemorewell-behaved
fermions [11]; (2) the Higgs boson is an approximate Goldstone boson, the only other known
mechanism forkeepinganelementaryscalarlight[12];(3)TheHiggsbosonisacompositescalar,
withaninverse lengthofordertheTeV-scale[12];(4)theEWSBscaleischosen bysomevacuum
selection mechanism [13]. Of course, maybe none of these explanations are relevant, and the
EWSB energy scale (which appears to us to be highly fine-tuned) is simply the result of some
initialcondition whoseoriginwillneverbediscernible.
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
In light of these remarks, how do we make further progress? We have at our disposal a very
successful experimental particle physics facility—the Large Hadron Collider (LHC), which has
only begun a comprehensive probe of the TeV-energy scale. Tothe experimentalists, Isay: “keep
searching fornewphysics beyond theSM(BSM).”Anyobserved departures fromSMpredictions
will contain critical clues to a more fundamental theory of elementary particles and their interac-
tions. Tothetheorists, Isay: “find new examples of BSMphysics (which might provide anatural
explanation to the EWSB scale) that may have been overlooked in LHC searches.” But what if
no signals for BSM physics emerge soon? My answer is: “look to the Higgs sector.” After all,
wehaveonlyrecently discovered amostremarkable particle thatseemstobelikenothing thathas
everbeen seen before—an elementary scalar boson. Shouldn’t weprobe this state thoroughly and
exploreitsproperties withasmuchprecision aspossible?
Putting considerations of naturalness aside, two critical questions to be addressed in future
LHCexperimentation are:
1. Arethereadditional Higgsbosons tobediscovered? (Thisincludes newcharged scalarsof
interest to this conference.) If fermionic matter and the gauge sector of the SM are non-minimal,
whyshouldn’t scalarmatteralsobenon-minimal? Toparaphrase I.I.Rabi,“whoorderedthat?”
2. If wemeasure the Higgs properties with sufficient precision, willdeviations from SM-like
Higgsbehavior berevealed?
Onemightbeconcerned thataddingadditional Higgsscalarstothetheorywillexacerbate the
fine-tuningproblemassociatedwiththeEWSBscale. Ofcourse,therearemanyexamplesinwhich
natural explanations of the EWSB scale employ BSM physics with extended Higgs sectors. The
minimalsupersymmetricextensionoftheSM(MSSM),whichemploystwoHiggsdoublets,isthe
most well known example of this type, but there are numerous other BSM examples as well. If
you give up on naturalness (e.g., vacuum selection), it has been argued that it may be difficult to
accommodate more than one Higgs doublet at the electroweak scale [14]. However, it is possible
toconstruct“partiallynatural”extendedHiggssectorsinwhichonescalarsquaredmassparameter
is fine-tuned (as in the SM), but additional scalar mass parameters are related to the EWSB scale
byasymmetry[15].
In the rest of this talk, I will take an agnostic approach and entertain the possibility of an
extended Higgs sector without providing a specific theoretical motivation. I shall focus on the
theoretical constraints on extended Higgs sectors in light of current experimental data (including
the fact that the observed Higgs boson is SM-like). These constraints will provide an important
frameworkforconsideringthephenomenologyofadditionalHiggsbosonsthatcouldbediscovered
infutureexperimentation attheLHC(oratfuturecolliderscurrently underconsideration).
2. Theoretical implicationsofa SM-like Higgsboson
BasedontheRun-ILHCHiggsdata[16],itisalreadyapparentthattheobservedHiggsboson
is SM-like. Thus any model of BSM physics, including models of extended Higgs sectors, must
incorporate this observation. In models of extended Higgs sectors, a SM-like Higgs boson can be
achieved inaparticular limitofthemodelcalledthealignmentlimit[17–21].
ConsideranextendedHiggssectorwithnhypercharge-oneHiggsdoubletsF andmadditional
i
singletHiggsfieldsf . Afterminimizingthescalarpotential,weassumethatonlytheneutralscalar
i
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
fields acquire vevs (in order to preserve electromagnetic charge conservation), F 0 =v/√2 and
h ii i
f 0 =x ,wherev2 (cid:229) v 2=4m2 /g2 (246GeV)2. Wedefinenewlinearcombinations,H,of
h ji j ≡ i| i| W ≃ i
thehypercharge-one doubletHiggsfields(thisistheso-called Higgsbasis[22–24]). Inparticular,
H = H1+ = 1(cid:229) v F , H0 =v/√2, (2.1)
1 H0 v ∗i i h 1i
1! i
and H ,H ,...,H are the other mutually orthogonal linear combinations of doublet scalar fields
2 3 n
such that H0 = 0 (for i = 1). That is H0 is aligned in field space with the direction of the
h i i 6 1
scalar field vev. In the alignment limit, h √2ReH0 v is a mass-eigenstate, and the tree-level
≡ 1 −
couplingsofhtoitself,togaugebosonsandtofermionsarepreciselythoseoftheSMHiggsboson.
In general, √2ReH0 v is not a mass-eigenstate due to mixing with other neutral scalars. Thus,
1 −
theobserved HiggsbosonisSM-likeifatleastoneofthefollowingtwoconditions aresatisfied:
1. Thediagonalsquaredmassesoftheotherscalarfieldsarealllargecomparedtothemassof
theobserved Higgsboson(theso-called decoupling limit[17,25]), and/or
2. The elements of the scalar squared mass matrix that govern the mixing of √2ReH0 v
1 −
withotherneutral scalarsaresuppressed.
IntheSM,m2=l v2wherev 246GeV,andl istheHiggsself-coupling[cf.eq.(1.1)]which
h ≃
shouldnotbemuchlargerthanO(1). Thus,weexpectm O(v). InextendedHiggssectors,there
h
∼
can be anew mass parameter, M v, such that allphysical Higgs masses with one exception are
≫
of O(M). The Higgs boson, with m O(v), is SM-like due to approximate alignment. This is
h
∼
the decoupling limit. After integrating out all the heavy degrees of freedom at the mass scale M,
oneisleftwithalow-energyeffectivetheorywhichconsistsoftheSMparticles,includingasingle
neutral scalarboson. Thislow-energy effectivetheoryisprecisely theSM!
Thealignmentlimitismostnaturallyachievedinthedecouplingregime. However,inthiscase
theadditionalHiggsbosonstatesareveryheavyandmaybedifficulttoobserveattheLHC.Inthe
case of approximate alignment without decoupling1 (due to suppressed scalar mixing), non-SM-
likeHiggsbosonstatesneednotbeveryheavyandthusaremoreeasilyaccessible attheLHC.
3. Examples ofextended Higgssectors near the alignmentlimit
3.1 ExtendingtheSMHiggssectorwithasingletscalar
ThesimplestexampleofanextendedHiggssectoraddsarealscalarfieldS. Themostgeneral
renormalizable gauge-invariant scalar potential (subject to a Z symmetry to eliminate linear and
2
cubictermsinS)is[29–31]
V(F ,S)= m2F †F m 2S2+1l (F †F )2+1l S4+l (F †F )S2. (3.1)
− − 2 1 2 2 3
Afterminimizing thescalarpotential, F 0 =v/√2and S =x/√2. Thesquared massmatrixof
h i h i
theneutralHiggsbosons is[32,33]
l v2 l vx
M2= 1 3 . (3.2)
l vx l x2
3 2 !
1In some models, alignment without decoupling can be achieved by a symmetry [26,27]. The inert doublet
model [28] isanoteworthy example inwhichtheexact alignment limitisaconsequence of adiscrete Z symmetry.
2
Inmostcases,approximatealignmentwithoutdecouplingisanaccidentalregionofthemodelparameterspace.
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
The corresponding mass eigenstates are h and H with m m . As discussed in Section 2, an
h H
≤
approximate alignment limitcan berealized intwodifferent ways: either l x vand/or x v.
3
| | ≪ ≫
In the case where l x v, h is SM-like if l v2 < l x2 and H is SM-like if l v2 > l x2. In
3 1 2 1 2
| | ≪
contrast, x vcorresponds tothedecoupling limit,wherehisSM-likeandm m .
H h
≫ ≫
TheHiggsmasseigenstates areexplicitly definedvia
h cosa sina √2ReF 0 v
= − − , (3.3)
H sina cosa √2S x
! ! !
−
where
l v2 = m2cos2a +m2 sin2a , (3.4)
1 h H
l x2 = m2sin2a +m2 cos2a , (3.5)
2 h H
l xv = (m2 m2)sina cosa . (3.6)
3 H h
−
TheSM-likeHiggsbosonisapproximately givenby√2ReF 0 v.
−
IfhisSM-like,thenm2 l v2 and
h≃ 1
l vx l vx
sina = | 3| | 3| 1, (3.7)
| | (m2H−m2h)(m2H−l 1v2) ≃ m2H−m2h ≪
q
IfH isSM-like,thenm2 l v2 and
H ≃ 1
l vx l vx
cosa = | 3| | 3| 1. (3.8)
| | (m2H−m2h)(l 1v2−m2h) ≃ m2H−m2h ≪
q
A phenomenological analysis presented in Ref. [33] shows that the allowed parameter regime
(consistent with the LHC Higgs data) roughly satisfies sina < 0.3 if m >m =125 GeV and
H h
| |
sina >0.9ifm <m =125GeV. ∼ ∼
h H
| |
∼
3.2 Thetwo-Higgsdoubletmodel(2HDM)
Considerthe2HDMwithhypercharge-one, doubletfieldsF andF [34,35]. Afterminimiz-
1 2
ingthescalarpotential, F 0 =v/√2(fori=1,2),wherev2+v2 (246GeV)2andtanb v /v .
h ii i 1 2≃ ≡ 2 1
TheHiggsbasisfieldsaredefinedas,
H+ v F +v F H+ v F +v F
H = 1 ∗1 1 ∗2 2, H = 2 − 2 1 1 2, (3.9)
1 H0 ≡ v 2 H0 ≡ v
1! 2!
suchthat H0 =v/√2and H0 =0. TheHiggsbasisisuniquelydefineduptoanoverallrephas-
h 1i h 2i
ingoftheHiggsbasisfieldH .
2
IntheHiggsbasis,thescalarpotential isgivenby[23,24,36]:
V =Y H†H +Y H†H +[Y H†H +h.c.]+1Z (H†H )2+1Z (H†H )2+Z (H†H )(H†H )
1 1 1 2 2 2 3 1 2 2 1 1 1 2 2 2 2 3 1 1 2 2
+Z (H†H )(H†H )+ 1Z (H†H )2+ Z (H†H )+Z (H†H ) H†H +h.c. , (3.10)
4 1 2 2 1 2 5 1 2 6 1 1 7 2 2 1 2
n o
where Y , Y and Z ,...,Z are real, whereas Y (cid:2), Z , Z and Z are pot(cid:3)entially complex. After
1 2 1 4 3 5 6 7
minimizingthescalarpotential,Y = 1Z v2 andY = 1Z v2.
1 −2 1 3 −2 6
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
For simplicity, we consider here the case of a CP-conserving scalar potential.2 In this case,
one can rephase the Higgs basis field H such that ImZ = ImZ = ImZ = 0. We identify the
2 5 6 7
CP-odd Higgs boson as A=√2Im H0, with m2 =Y +1(Z +Z Z )v2. After eliminating Y
2 A 2 2 3 4− 5 2
in favor of m2, the CP-even Higgs squared mass matrix with respect to the Higgs basis states,
A
√2ReH0 v,√2ReH0 ,isgivenby
{ 1− 2}
Z v2 Z v2
M2 = 1 6 . (3.11)
H Z v2 m2+Z v2
6 A 5 !
The CP-even Higgs bosons are h and H with m m . The couplings of √2ReH0 v coincide
h ≤ H 1 −
withthoseoftheSMHiggsboson.3 Thus,thealignment limitcorresponds totwolimitingcases:
1. Z 1. Inthiscase,hisSM-likeifm2+(Z Z )v2 >0;otherwise, H isSM-like.
| 6|≪ A 5− 1
2. m2 (Z Z )v2. Thisisthedecoupling limit;hisSM-likeandm m m m .
A≫ 1− 5 A∼ H ∼ H± ≫ h
Inparticular, theCP-evenmasseigenstates are:
Hh!= csbb −aa −csbb −aa ! √√2 2ReRHeH10−20 v!, (3.12)
− −
wherecb a cos(b a )andsb a sin(b a )aredefinedintermsoftheanglea thatdiagonal-
− ≡ − − ≡ −
izes the CP-even Higgs squared mass matrix when expressed in the original basis of scalar fields,
√2ReF 0 v ,√2ReF 0 v , and tanb v /v . Since the SM-like Higgs boson is approxi-
{ 1− 1 2− 2} ≡ 2 1
mately√2ReH10−v,itfollowsthathisSM-likeif|cb −a |≪1,andH isSM-likeif|sb −a |≪1.
The approximate alignment limit can be derived more explicitly as follows. The CP-even
Higgssquared massmatrixyields[38,39]
Z v2 = m2s2 +m2c2 ,
1 h b a H b a
− −
Z6v2 = (m2h m2H)sb a cb a ,
− − −
Z v2 = m2s2 +m2c2 m2.
5 H b a h b a A
−
− −
IfhisSM-like,thenm2 Z v2 and
h≃ 1
Z v2 Z v2
6 6
cb a = | | | | 1. (3.13)
| − | (m2H−m2h)(m2H−Z1v2) ≃ m2H−m2h ≪
q
The decoupling limit is realized when m m . In contrast, alignment without decoupling re-
H h
≫
quiresthat Z 1andm O(v). IfH isSM-like,thenm2 Z v2 and[40]
| 6|≪ H ∼ H ≃ 1
Z v2 Z v2
6 6
sb a = | | | | 1, (3.14)
| − | (m2H−m2h)(Z1v2−m2h) ≃ m2H−m2h ≪
q
which can only be achieved if Z 1. In particular a SM-like H can only arise in the limit of
6
| |≪
alignment withoutdecoupling.
2Themoregeneralcaseinwhichnoscalarbasisexistssuchthatalltheparametersofeq.(3.10)aresimultaneously
realistreatedinRefs.[36,37].
3Although the tree-level couplings of √2ReH0 v coincide with those of the SM Higgs boson, the one-loop
1 −
couplingscandifferduetotheexchangeofnon-minimalHiggsstates(ifnottooheavy).Forexample,thechargedHiggs
bosonloopinterfereswiththeW andfermionloopcontributionstotheamplitudeforthedecayoftheSM-likeHiggs
bosontogg orgZ.
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
Sofar,wehavenotyetdiscussedthecouplingsoftheHiggsbosonstofermions. IntheF –F
1 2
basis, the2HDMHiggs-quark YukawaLagrangian is[37]:
L =U F 0 hUU D K†F hUU +U KF +hD†D +D F 0hD†D +h.c., (3.15)
− Y L i∗ i R− L −i i R L i i R L i i R
where K is the CKM mixing matrix, hU,D are 3 3 Yukawa coupling matrices, and there is an
×
implicit sum over i =1,2. Unlike in the SM, the diagonalization of the quark masses does not
automatically diagonalize the neutral-Higgs–quark Yukawa coupling matrices. Hence, the gen-
eral 2HDM possesses tree-level Higgs-mediated flavor-changing neutral currents (FCNCs) which
are generically too large and thus inconsistent with experimental data. In order to naturally elim-
inate tree-level Higgs-mediated FCNC [41,42], one can impose a discrete symmetry to restrict
the structure of L .Twodifferent choices for how the discrete symmetry acts on the quarks then
Y
yield: Type-IYukawacouplings [43,44]ifhU =hD =0,andType-IIYukawacouplings [44,45]if
1 1
hU =hD=0. (Similarconsiderations canalsobeappliedtotheHiggs-lepton Yukawacouplings.)
1 2
ItisstraightforwardtoworkouttheHiggscouplingsintheapproximatealignmentlimit. Some
examples are provided in Table 1 in the case where h is SM-like with a CP-conserving scalar
potential and Type-I or II Yukawa couplings. In the third column of Table 1, the first non-trivial
corrections to the alignment limit are presented. Note that these corrections are correlated, and
the approach to decoupling is governed by cb a . Thus, any deviations from SM-like behavior
−
of the observed Higgs boson can provide important clues to the structure of the extended Higgs
sector. Thephenomenology ofthe 2HDMintheapproximate alignment limit andits implications
for future LHC experimental studies have recently been elucidated in Refs. [39,40] in the case
wherehorH,respectively, isidentifiedastheobserved Higgsbosonofmass125GeV.
Higgsinteraction 2HDMcoupling approach toalignmentlimit
hVV sb −a 1−12cb2−a
hhh * 1+2(Z6/Z1)cb a
−
hH+H− * 13 (Z3/Z1)+(Z7/Z1)cb a
−
Hhh * −Z6/(cid:2)Z1+ 1−23(Z345/Z1) cb(cid:3)−a
hhhh * 1+(cid:2)3(Z6/Z1)cb a (cid:3)
−
hDD sb a 1+cb a r RD 1+cb a r RD
− − −
hUU sb a 1+cb a r RU 1+cb a r RU
− − −
Table1: The2HDMcouplingsoftheSM-likeHiggsbosonhnormalizedtothoseoftheSMHiggsboson,
intheapproachtothealignmentlimit. ThehH+H andHhhcouplingsarenormalizedtotheSMhhhcou-
−
pling [17,39] (where Z Z +Z +Z ). The scalar Higgs potentialis taken to be CP-conserving. For
345 3 4 5
≡
theHiggscouplingstofermions,Disacolumnvectorofthreedown-typefermionfields(eitherdown-type
quarksorchargedleptons)andU isacolumnvectorofthreeup-typequarkfields. ForType-IYukawacou-
plings,r D=r U =1cotb ,andforType-IIYukawacouplings,r D= 1tanb andr U =1cotb [19].Inthe
R R R − R
thirdcolumnabove,thefirstnon-trivialcorrectiontoalignmentisexhibited. Finally,completeexpressions
fortheentriesmarkedwitha*canbefoundinRefs.[36,37].
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
3.3 TheMSSMHiggssector
TheMSSMHiggssectorisaCP-conservingType-II2HDM.Thedimension-four termsofthe
scalarpotentialareconstrained bysupersymmetry. Inparticular, thescalarpotentialparameters(at
treelevel)intheHiggsbasisaredetermined bytheelectroweak gaugecouplings [36],
Z1 = Z2= 41(g2+g′2)c22b , Z5= 14(g2+g′2)s22b , Z7=−Z6= 14(g2+g′2)s2b c2b ,
Z = Z +1(g2 g2), Z =Z 1g2, (3.16)
3 5 4 − ′ 4 5−2
in a convention where tanb 0, where c2b cos2b and s2b sin2b . It then follows from
≥ ≡ ≡
eq.(3.13)that,
m4s2 c2
cos2(b a )= Z 2b 2b . (3.17)
− (m2 m2)(m2 m2c2 )
H− h H− Z 2b
Thedecouplinglimitisachievedwhenm m asexpected. Exactalignmentwithoutdecoupling
H h
≫
is (naively) possible at tree-level when Z =0, which yields sin4b =0 and m2 =Z v2 =m2c2 .
6 h 1 Z 2b
However,theseresultsareinconsistent withtheobserved Higgsmassof125GeV.
Itiswellknownthatradiativecorrectionscansignificantlymodifythetree-levelHiggsproper-
ties[46]. Inparticular, considerthelimitwherem ,m ,m ,m M ,whereM2 m m isthe
h A H H± ≪ S S ≡ t˜1 t˜2
productoftopsquarkmasses. Inthiscase,onecanformallyintegrateoutthesquarksandgenerate
a low-energy effective 2HDM Lagrangian (which is no longer of the tree-level MSSM form). At
one-loop, thedominantcontributions totheeffectiveZ andZ parameters aregivenby[47]4
1 6
Z v2 = m2c2 +3v2sb4ht4 ln MS2 + Xt2 1 Xt2 , (3.18)
1 Z 2b 8p 2 m2 M2 −12M2
(cid:20) (cid:18) t (cid:19) S (cid:18) S(cid:19)(cid:21)
Z6v2 = −s2b m2Zc2b −3v126spb22ht4 ln Mm2S2 +Xt(X2Mt+2Yt)−1X2t3MYt4 , (3.19)
( (cid:20) (cid:18) t (cid:19) S S(cid:21))
wheresb sinb ,ht isthetopquarkYukawacoupling, Xt At m cotb andYt At+m tanb .
≡ ≡ − ≡
Notethatm2 Z v2isconsistentwithm 125GeVforsuitablechoicesforM andX. Exact
h≃ 1 h≃ S t
alignment (i.e., Z =0) can now be achieved due to an accidental cancellation between tree-level
6
andloopcontributions [47],
m2Zc2b = 3v126spb22ht4 ln Mm2S2 +Xt(X2Mt+2Yt)−1X2t3MYt4 . (3.20)
(cid:20) (cid:18) t (cid:19) S S(cid:21)
Onecanmanipulateeq.(3.20)intoa7thorderpolynomialequationintanb . Thealignmentcondi-
tionisthenachievedby(numerically)solvingthisequationforpositiverealsolutionsoftanb . Fol-
lowingarecipeprovidedbyRefs.[48,49],onecanfurtherimproveeqs.(3.18)and(3.19)toinclude
theleadingtwo-loopcorrectionsofO(a h2)byreplacingh withh (l ),wherel m (m )M 1/2
s t t t t t S
≡
intheone-loop leading logcontributions andl M intheleading threshold corrections. Impos-
≡ S (cid:2) (cid:3)
ing Z =0now leads toa11th order polynomial equation intanb thatcan besolved numerically.
6
Threepositivesolutions areexhibited inFig.1asafunctionof m /M andA /M [50].
S t S
4CP-violatingphases,whichcouldappearintheMSSMparameterssuchasm andAt,areneglected.
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FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
Figure1: Contoursoftanb correspondingtoexactalignment,Z =0, inthe(m /M ,A/M )plane. Z is
6 S t S 1
adjusted to givethe correctHiggsmass. Theleadingone-loopandtwo-loopcorrectionsofO(a h2) to Z
s t 1
and Z have been included. Taking the three panels together, one can immediately discern the regionsof
6
zero,one,twoandthreevaluesoftanb inwhichexactalignmentisrealized.TakenfromRef.[50].
In light of the SM-like nature of the observed Higgs boson, the ATLAS Collaboration con-
cludedthatm >370GeV[51]. However,thisanalysisfailedtoconsiderthepossibilityofapprox-
A
∼
imate alignment without decoupling [47], which can be achieved in certain regions of the MSSM
parameter space [50]. The direct searches for H and A (decaying into t +t ) by the ATLAS and
−
CMS Collaborations in the mass region from 200–370 GeV provide no constraints for values of
tanb <8–10[52,53]. Inthisparameterregime,approximatealignmentisstillpossibleforsuitable
choice∼s of m /M andA /M . Arecent pMSSMparameter scan [50], which takes into account the
S t S
observed Higgs data, direct searches for H and A, indirect constraints from heavy flavor physics,
and supersymmetric particle searches, finds that values of m as low as 200 GeV are within 2s
A
of the best fit point obtained by a global likelihood analysis, under the assumption that h is the
observed Higgsbosonofmass125GeV.5
Alignment without decoupling can also be achieved in the NMSSM (where an additional
Higgssinglet superfieldisaddedtotheMSSM).Forfurtherdetails, seeRef.[54].
3.4 BeyondHiggssingletsanddoublets
Ifoneconsiders ascalarsectorwithtripletHiggsfields,thenonemustalsoinclude additional
Higgs multiplets in such a way that the electroweak r -parameter is approximately equal to 1.
GeorgiandMachacekconstructed amodelinwhichr =1attree-levelduetoawellchosenscalar
potentialthatrespectsthecustodialsymmetry[55]. TheirmodelcontainsacomplexY =1doublet,
a complexY =2 triplet and a realY =0 singlet. After minimizing the scalar potential, there is a
doublet vev,vf ,andacommontripletvev,vc ,withv2 vf2 +8v2c (246GeV)2.
≡ ≃
ThephysicalscalarsmakeupcustodialSU(2)multiplets: a5-pletofstates(H ,H andH0)
5±± 5± 5
with common massm , atriplet (H , H0)with common mass m , and custodial singlets that mix
5 3± 3 3
withsquaredmassmatrix[56]
M2= Z11v2f vf vc (Z12−2√3m23/v2) , (3.21)
vf vc (Z12−2√3m23/v2) 32m23−12m25+v2c (Z22−12m23/v2)!
wheretheZ dependonthedimensionlessquarticcouplings. ThecustodialsingletCP-evenHiggs
ij
bosons are h and H with m m . An approximate alignment limit can be realized in two dif-
h H
≤
5ThesameanalysisalsoyieldsanallowedparameterregimewhereHistheobservedHiggsbosonofmass125GeV.
8
FutureHiggsStudies: ATheorist’sOutlook HowardE.Haber
ferent ways. First, if vc ≪v, then h is SM-like if Z11v2 < 23m23− 12m25; otherwise, H is SM-like.
Alternatively, inthedecoupling limit,hisSM-likeandm m m m [56].
H 3 5 h
≃ ≃ ≫
OneinterestingfeatureoftheGeorgi-Machacek modelisthattheexistenceofdoubly-charged
Higgsbosons modifiestheunitarity sumrule[57],
(cid:229) g2 =g2m2 +(cid:229) g 2, (3.22)
hiW+W− W | Hk++W−W−|
i k
where the sum is taken over all CP-even Higgs bosons of the model. The presence on the last
term on the right hand side of eq. (3.22) means that individual hVV couplings can exceed the
i
correspondingSMHiggscouplingtoVV. ItisconvenienttowritecH cosq H =vf /(v2f +8v2c )1/2,
≡
ands sinq . Then,thefollowingcouplings arenoteworthy[58]:
H H
≡
H0W+W : gc m , H 0W+W : 8/3gm s ,
1 − H W 1′ − W H
H50W+W−: 1/3gmWsH, H5++W−W−: p√2gmWsH,
whereH0andH 0arethecustodipalsingletinteractioneigenstates. AmongthefourHiggsstatesthat
1 1′
couple toWW,whosecouplings arelistedabove,H 0,H0 andH++ havenocoupling tofermions,
1′ 5 5
whereastheH0f f¯coupling is gm /(2m c ).
1 − q W H
Ingeneral H0 and H 0 canmix. Intheabsence ofH0–H 0 mixing, c =1corresponds tothe
1 1′ 1 1′ H
alignment limit. But consider the strange case of s = 3/8, where the H 0 coupling toW+W
H 1′ −
matches that of the SM. Nevertheless, this does not saturate the HWW sum rule! Moreover, it is
p
possible that the H 0W+W coupling is larger than the SM value of gm , without violating the
1′ − W
sum rule given by eq. (3.22). Including H0–H 0 mixing allows for even more baroque scenarios
1 1′
thatarenotpossible inamulti-doublet extension oftheSMHiggssector.
4. Conclusions
Given the non-minimal nature of the observed spectrum of fundamental fermions and gauge
bosons, itwouldberemarkable iftheHiggsboson wereasoloact. Thus,thesearch foradditional
scalars that exist in an extended Higgs sector will be an important enterprise in the experimental
program attheLHCandatanyfuturecolliderfacility.
The current Higgs data strongly suggest that the observed Higgs boson is SM-like. This al-
ready places a strong constraint on the theoretical structure of any non-minimal Higgs sector. In
particular, the alignment limit, in which the mass eigenstate corresponding to the observed Higgs
bosonisalignedwiththedirection(infieldspace)ofthescalardoubletvev,mustbeagoodapprox-
imation. Thesimplest waytoachieve thealignment limitisinthecasewherealladditional Higgs
scalars are significantly heavier than the observed Higgs boson (corresponding to the decoupling
limit). But, we have also argued for the possibility of the alignment limit without decoupling if
themixingbetween theSMHiggsbosonandtheadditional neutral Higgsscalars issuppressed, in
whichcaseallHiggsscalarsmaybelight[ofO(v)]andthusmoreaccessible toLHCsearches.
Finally, as the Higgs data become more precise, deviations from SM properties of the Higgs
boson may eventually be observed. Indeed, departures from the alignment limit encode critical
information that can provide important clues for the structure of the non-minimal Higgs sector.
PursuingHiggsphysicsintothefuturebytheoristsandexperimentalistsislikelytoleadtoprofound
insights intothefundamental theoryofparticlesandtheirinteractions.
9